Kinetic and dynamic data structures for closest pair and all nearest neighbors

We present simple, fully dynamic and kinetic data structures, which are variants of a dynamic two-dimensional range tree, for maintaining the closest pair and all nearest neighbors for a set of n moving points in the plane; insertions and deletions of points are also allowed. If no insertions or deletions take place, the structure for the closest pair uses O(n log n) space, and processes O(n2 s+2(n) log n) critical events, each in O(log2 n) time. Here s is the maximum number of times where the distances between any two specific pairs of points can become equal, s(q) = s(q)/q, and s(q) is the maximum length of Davenport-Schinzel sequences of order s on q symbols. The dynamic version of the problem incurs a slight degradation in performance: If m n insertions and deletions are performed, the structure still uses O(n log n) space, and processes O(mns+2(n) log3 n) events, each in O(log3 n) time. Our kinetic data structure for all nearest neighbors uses O(n log2 n) space, and processes O(n...

• Type: Journal
• Year: 2008
• Where: TALG
• Authors: Pankaj K. Agarwal, Haim Kaplan, Micha Sharir
• Keywords: Closest Pair, Kinetic Data Structure, Nearest Neighbors, TALG 2008